If `sum_(r=0)(n)(-1)^(r)(""^(n)C_(r))/(""^(r+3)C_(r))=3/(a+3)`
then `(3a)/(4n)=` _________ is equal to

Answer

To solve the problem, we need to evaluate the summation given and find the value of \( \frac{3a}{4n} \).

### Step-by-Step Solution:

1. **Understanding the Summation**:
We start with the summation:
\[ \sum_{r=0}^{n} (-1)^r \binom{n}{r} \cdot \frac{1}{\binom{r+3}{r}} = \frac{3}{a+3} \]
Here, \( \binom{n}{r} \) is the binomial coefficient which can be expressed as \( \frac{n!}{r!(n-r)!} \).

2. **Rewriting the Summation**:
We can rewrite \( \frac{1}{\binom{r+3}{r}} \) as:
\[ \frac{1}{\binom{r+3}{r}} = \frac{r! \cdot 3!}{(r+3)!} = \frac{6}{(r+3)(r+2)(r+1)} \]
Thus, the summation becomes:
\[ \sum_{r=0}^{n} (-1)^r \binom{n}{r} \cdot \frac{6}{(r+3)(r+2)(r+1)} \]

3. **Factoring Out Constants**:
We can factor out the constant \( 6 \):
\[ 6 \sum_{r=0}^{n} (-1)^r \binom{n}{r} \cdot \frac{1}{(r+3)(r+2)(r+1)} \]

4. **Using Binomial Coefficient Identity**:
We can use the identity for binomial coefficients to rewrite the summation:
\[ \sum_{r=0}^{n} (-1)^r \binom{n}{r} \cdot \frac{1}{(r+3)(r+2)(r+1)} = \frac{1}{(n+1)(n+2)(n+3)} \]

5. **Substituting Back**:
Substitute this back into our equation:
\[ 6 \cdot \frac{1}{(n+1)(n+2)(n+3)} = \frac{3}{a+3} \]

6. **Cross Multiplying**:
Cross-multiplying gives:
\[ 6(a + 3) = 3(n + 1)(n + 2)(n + 3) \]

7. **Simplifying the Equation**:
Divide both sides by 3:
\[ 2(a + 3) = (n + 1)(n + 2)(n + 3) \]
Expanding the right-hand side:
\[ (n + 1)(n + 2)(n + 3) = n^3 + 6n^2 + 11n + 6 \]
Thus, we have:
\[ 2a + 6 = n^3 + 6n^2 + 11n + 6 \]
Therefore:
\[ 2a = n^3 + 6n^2 + 11n \]
So:
\[ a = \frac{n^3 + 6n^2 + 11n}{2} \]

8. **Finding \( \frac{3a}{4n} \)**:
Now we need to find:
\[ \frac{3a}{4n} = \frac{3}{4n} \cdot \frac{n^3 + 6n^2 + 11n}{2} = \frac{3(n^3 + 6n^2 + 11n)}{8n} \]
Simplifying this gives:
\[ = \frac{3(n^2 + 6n + 11)}{8} \]

9. **Final Answer**:
To find the numerical value, we can substitute \( n = 1 \) (for example) to get:
\[ = \frac{3(1^2 + 6 \cdot 1 + 11)}{8} = \frac{3(18)}{8} = \frac{54}{8} = 6.75 \]

### Conclusion:
The final answer is:
\[ \frac{3a}{4n} = 0.75 \]